Difference between revisions of "2006 SMT/Geometry Problems/Problem 3"
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Latest revision as of 12:22, 28 May 2012
Problem
Circle 
is centered at 
with radius 
. Circle 
is externally tangent to circle and tangent to the 
axis. Find an equation, solved for 
if possible, for the locus of possible centers 
of circle .
Solution
![[asy] unitsize(1cm); draw((-5,0)--(5,0)); draw((0,0)--(0,5)); draw(circle((0,3),1)); dot((0,3)); draw(circle((2,3/2),3/2)); dot((2,3/2)); draw((2,3/2)--(2,0)); draw((2,3/2)--(0,3)); label("$\gamma$",(-.5,4),W); label("$\delta$",(7/2,3/2),E); label("$(x,y)$",(2,3/2),ENE); label("$(0,3)$",(0,3),NE); [/asy]](http://latex.artofproblemsolving.com/e/7/6/e768e4776490a89659cf24ae53d38864dffaef0e.png)
For to be tangent to both the axis and , its distance from must be equal to its distance from the axis. Note that its distance from the axis is just , and its distance from is equal its distance from 
minus the radius, which is . Therefore, by the distance formula,
![\[\sqrt{(x-0)^2+(y-3)^2}-1=y\]](http://latex.artofproblemsolving.com/c/b/5/cb51df35b1c3e441193c0afb61de7be6f53bb4ad.png)
![\[x^2+y^2-6y+9=(y+1)^2=y^2+2y+1\]](http://latex.artofproblemsolving.com/a/5/e/a5eba09b333d5d1ae76e4a4296006691561ae317.png)
![\[8y=x^2+8\]](http://latex.artofproblemsolving.com/0/8/1/08135003c8958f7ce277c40ff273c913e55b0a05.png)
![\[\boxed{y=\frac{x^2}{8}+1}\]](http://latex.artofproblemsolving.com/f/5/2/f524195dae826309940c4b3d8299ac145ad022a6.png)
